Subgroups — C₂×#11746898 #14437894

Order 1920.

872 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4021	C₂	1	1	yes	maximal, characteristic
3	C₂×#91087	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90970	C₂	1	2	yes	maximal, complemented
5	C₂×#90964	C₂	1	2	yes	maximal
6	#11746898	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65378	—	3	1	no	maximal, Hall
8	C₂²×C₅×Dic₆	—	2	1	no	—
9	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
10	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊#168	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊Dic₃	—	2	1	no	—
14	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₅×#869	—	2	2	no	—
16	C₅×#869	C₂²	1	4	yes	complemented
17	C₂×C₃×#685	—	2	1	no	—
18	C₂×C₃×#685	—	2	1	no	—
19	C₃×#4021	—	2	2	no	—
20	C₃×#4021	C₂²	1	4	yes	—
21	#91087	C₂²	1	4	yes	complemented
22	#90970	C₂²	1	8	yes	complemented
23	#90964	C₂²	1	8	yes	—
24	C₂×#5055	—	5	1	no	maximal, Hall
25	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
26	C₂×#4021	D₃	1	1	yes	characteristic
27	#65378	—	3	8	no	—
28	C₂×#3942	—	3	2	no	—
29	C₂×#3941	—	3	2	no	—
30	C₂×#3935	—	3	1	no	—
31	C₂³×C₃×D₅	—	2	1	no	—
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₅×Dic₃	—	2	2	no	—
36	C₂×C₅×Dic₆	—	2	4	no	—
37	C₂×C₅×Dic₆	—	4	4	no	—
38	C₂²×C₃×Dic₅	—	2	1	no	—
39	C₂×C₃×D₂₀	—	2	4	no	—
40	C₂×C₃×D₂₀	—	2	1	no	—
41	C₂×C₃×D₂₀	C₂³	1	2	yes	—
42	C₂×C₃×Dic₁₀	—	2	4	no	—
43	C₂×C₃×Dic₁₀	—	2	1	no	—
44	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
45	C₃×C₄²×C₅	D₄	1	2	yes	—
46	C₃×C₄²×C₅	C₂³	1	2	yes	—
47	C₂×C₃⋊#168	—	2	4	no	—
48	C₂×C₃⋊#168	C₂³	1	4	yes	—
49	C₅×C₄⋊Dic₃	—	2	2	no	—
50	C₅×C₄⋊Dic₃	—	2	4	no	—
51	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
52	C₃×#685	—	2	4	no	—
53	C₃×#685	—	2	4	no	—
54	C₂×C₃×#124	—	5	1	no	—
55	C₂×#869	D₅	1	1	yes	characteristic, complemented
56	C₂×#835	—	5	2	no	—
57	C₂×#832	—	5	2	no	—
58	#5055	—	5	8	no	—
59	C₂×#803	—	5	1	no	—
60	C₂²×C₅×Q₈	—	6	1	no	—
61	C₂²×D₂₀	C₂×D₃	1	1	yes	characteristic
62	C₂²×Dic₁₀	C₂×D₃	1	1	yes	characteristic
63	C₅×C₄⋊Q₈	—	3	4	no	—
64	C₅×C₄⋊Q₈	—	6	2	no	—
65	C₂×C₅×C₄⋊C₄	—	3	2	no	—
66	C₂×C₅×C₄⋊C₄	—	6	1	no	—
67	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
68	C₂×#685	—	2	1	no	—
69	C₂×#685	—	2	1	no	—
70	C₂²×#168	—	3	2	no	—
71	#4021	—	2	2	no	—
72	#4021	C₂×D₃	1	4	yes	—
73	#3942	—	3	8	no	—
74	#3941	—	3	8	no	—
75	#3935	—	3	4	no	—
76	C₂×#489	—	15	1	no	Sylow
77	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×D₅	—	2	2	no	—
79	C₂²×C₃×D₅	—	4	2	no	—
80	C₂²×C₃×D₅	—	4	2	no	—
81	C₂²×C₃×D₅	—	4	2	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	4	yes	—
88	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
89	C₂×C₅×Dic₃	—	2	4	no	—
90	C₂×C₅×Dic₃	—	4	4	no	—
91	C₅×Dic₆	—	4	8	no	—
92	C₂×C₃×Dic₅	—	2	2	no	—
93	C₂×C₃×Dic₅	—	4	2	no	—
94	C₃×D₂₀	—	2	4	no	—
95	C₃×D₂₀	—	4	2	no	—
96	C₃×Dic₁₀	—	2	4	no	—
97	C₃×Dic₁₀	—	4	2	no	—
98	C₃⋊#168	—	2	4	no	—
99	C₂²×C₃×Q₈	—	5	1	no	—
100	C₂²×C₃×D₄	—	5	1	no	—
101	C₂²×Dic₆	—	2	1	no	—
102	C₃×#124	—	5	4	no	—
103	C₃×#124	—	10	2	no	—
104	C₂×C₃×C₂²⋊C₄	—	10	1	no	—
105	C₂×C₃×C₂²⋊C₄	—	10	1	no	—
106	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
107	C₂×C₄⋊Dic₃	—	2	1	no	—
108	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
109	C₂²×C₃⋊C₈	—	5	2	no	—
110	#869	—	2	2	no	—
111	#869	C₂×D₅	1	4	yes	complemented
112	#835	—	5	8	no	—
113	#832	—	5	8	no	—
114	#803	—	5	4	no	—
115	C₂³×D₅	—	2	1	no	—
116	C₂×C₅×Q₈	—	6	4	no	—
117	C₂×C₅×Q₈	—	12	4	no	—
118	C₂²×C₄×C₅	—	6	2	no	—
119	C₂²×C₄×C₅	#67	1	1	yes	characteristic
120	C₂²×C₄×C₅	#67	1	1	yes	characteristic
121	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
122	C₂²×Dic₅	—	2	1	no	—
123	C₂×D₂₀	—	2	4	no	—
124	C₂×D₂₀	—	2	1	no	—
125	C₂×D₂₀	C₂²×D₃	1	2	yes	—
126	C₂×Dic₁₀	—	2	4	no	—
127	C₂×Dic₁₀	—	2	1	no	—
128	C₂×Dic₁₀	C₂²×D₃	1	2	yes	—
129	C₅×C₄⋊C₄	—	3	4	no	—
130	C₅×C₄⋊C₄	—	6	2	no	—
131	C₅×C₄⋊C₄	—	6	4	no	—
132	C₄²×C₅	#67	1	2	yes	—
133	C₄²×C₅	C₂²×D₃	1	2	yes	—
134	#685	—	2	4	no	—
135	#685	—	2	4	no	—
136	C₂×#168	—	3	4	no	—
137	C₂×#168	—	6	4	no	—
138	C₂×C₄⋊Q₈	—	3	1	no	—
139	C₂×#124	—	5	1	no	—
140	#489	—	15	8	no	—
141	C₂×#103	—	15	2	no	—
142	C₂×#102	—	15	2	no	—
143	C₂×#97	—	15	1	no	—
144	C₂²×C₃×C₅	#124	1	2	yes	—
145	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
146	C₂²×C₃×C₅	#137	1	2	yes	—
147	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₃×D₅	—	4	2	no	—
149	C₂×C₃×D₅	—	4	2	no	—
150	C₂×C₃×D₅	—	4	2	no	—
151	C₂×C₃×D₅	—	8	4	no	—
152	C₃×C₄×C₅	—	2	2	no	—
153	C₃×C₄×C₅	—	2	2	no	—
154	C₃×C₄×C₅	#124	1	2	yes	—
155	C₃×C₄×C₅	C₂×#41	1	2	yes	—
156	C₃×Dic₅	—	4	2	no	—
157	C₅×Dic₃	—	4	4	no	—
158	C₂⁴×C₃	—	10	1	no	—
159	C₂×C₃×Q₈	—	5	2	no	—
160	C₂×C₃×Q₈	—	10	1	no	—
161	C₂×C₃×Q₈	—	10	4	no	—
162	C₂×C₃×D₄	—	5	2	no	—
163	C₂×C₃×D₄	—	10	4	no	—
164	C₂×C₃×D₄	—	10	1	no	—
165	C₂²×C₃×C₄	—	10	1	no	—
166	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
167	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
168	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
169	C₂²×Dic₃	—	2	2	no	—
170	C₂×Dic₆	—	2	4	no	—
171	C₂×Dic₆	—	4	4	no	—
172	C₃×C₂²⋊C₄	—	10	4	no	—
173	C₃×C₂²⋊C₄	—	10	4	no	—
174	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
175	C₃×C₄²	C₂²×D₅	1	2	yes	—
176	C₄⋊Dic₃	—	2	2	no	—
177	C₄⋊Dic₃	—	2	4	no	—
178	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
179	C₂×C₃⋊C₈	—	5	4	no	—
180	C₂×C₃⋊C₈	—	10	4	no	—
181	C₂³×C₅	C₂×#67	1	1	yes	characteristic
182	C₂²×D₅	—	2	2	no	—
183	C₂²×D₅	—	4	2	no	—
184	C₂²×D₅	—	4	2	no	—
185	C₂²×D₅	—	4	2	no	—
186	C₅×Q₈	—	12	8	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	6	4	no	—
190	C₂×C₄×C₅	—	12	4	no	—
191	C₂×C₄×C₅	#237	1	4	yes	—
192	C₂×C₄×C₅	C₂×#67	1	2	yes	—
193	C₂×C₄×C₅	C₂×#67	1	2	yes	—
194	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
196	C₂×Dic₅	—	2	2	no	—
197	C₂×Dic₅	—	4	2	no	—
198	D₂₀	—	2	4	no	—
199	D₂₀	—	4	2	no	—
200	Dic₁₀	—	2	4	no	—
201	Dic₁₀	—	4	2	no	—
202	#168	—	6	4	no	—
203	C₂²×Q₈	—	5	1	no	—
204	C₂²×Q₈	—	6	1	no	—
205	C₂²×D₄	—	5	1	no	—
206	C₂²×C₈	—	15	2	no	—
207	C₄⋊Q₈	—	3	4	no	—
208	C₄⋊Q₈	—	6	2	no	—
209	#124	—	5	4	no	—
210	#124	—	10	2	no	—
211	C₂×C₄⋊C₄	—	3	2	no	—
212	C₂×C₄⋊C₄	—	6	1	no	—
213	C₂×C₂²⋊C₄	—	10	1	no	—
214	C₂×C₂²⋊C₄	—	10	1	no	—
215	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
216	#103	—	15	8	no	—
217	#102	—	15	8	no	—
218	#97	—	15	4	no	—
219	C₂×C₃×C₅	#489	1	4	yes	complemented
220	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
221	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂×C₃×C₅	C₂×#137	1	1	yes	characteristic
223	C₃×D₅	—	8	2	no	—
224	C₂³×C₃	—	10	2	no	—
225	C₂³×C₃	—	20	2	no	—
226	C₂³×C₃	—	20	2	no	—
227	C₂³×C₃	—	20	2	no	—
228	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
229	C₃×Q₈	—	10	4	no	—
230	C₃×Q₈	—	20	2	no	—
231	C₃×D₄	—	10	4	no	—
232	C₃×D₄	—	20	2	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	2	2	no	—
235	C₂×C₃×C₄	—	10	2	no	—
236	C₂×C₃×C₄	—	20	2	no	—
237	C₂×C₃×C₄	#713	1	4	yes	—
238	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
239	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
240	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
241	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
242	C₂×Dic₃	—	2	4	no	—
243	C₂×Dic₃	—	4	4	no	—
244	Dic₆	—	4	8	no	—
245	C₃⋊C₈	—	10	4	no	—
246	C₂²×C₅	#936	1	2	yes	—
247	C₂²×C₅	#949	1	2	yes	—
248	C₂²×C₅	#951	1	2	yes	—
249	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
250	C₂×D₅	—	4	2	no	—
251	C₂×D₅	—	4	2	no	—
252	C₂×D₅	—	4	2	no	—
253	C₂×D₅	—	8	4	no	—
254	C₄×C₅	—	2	2	no	—
255	C₄×C₅	—	2	2	no	—
256	C₄×C₅	—	12	4	no	—
257	C₄×C₅	#936	1	2	yes	—
258	C₄×C₅	C₂×#237	1	2	yes	—
259	Dic₅	—	4	2	no	—
260	C₂⁴	—	10	1	no	—
261	C₂×Q₈	—	5	2	no	—
262	C₂×Q₈	—	6	4	no	—
263	C₂×Q₈	—	10	1	no	—
264	C₂×Q₈	—	10	4	no	—
265	C₂×Q₈	—	12	4	no	—
266	C₂×D₄	—	5	2	no	—
267	C₂×D₄	—	10	4	no	—
268	C₂×D₄	—	10	1	no	—
269	C₂²×C₄	—	6	2	no	—
270	C₂²×C₄	—	10	1	no	—
271	C₂²×C₄	#1201	1	1	yes	characteristic
272	C₂²×C₄	#1201	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
274	C₂×C₈	—	15	4	no	—
275	C₂×C₈	—	30	4	no	—
276	C₄⋊C₄	—	3	4	no	—
277	C₄⋊C₄	—	6	2	no	—
278	C₄⋊C₄	—	6	4	no	—
279	C₂²⋊C₄	—	10	4	no	—
280	C₂²⋊C₄	—	10	4	no	—
281	C₄²	#1201	1	2	yes	—
282	C₄²	C₂×D₃×D₅	1	2	yes	—
283	C₃×C₅	C₂×#489	1	1	yes	characteristic, complemented, Hall
284	C₂²×C₃	—	20	2	no	—
285	C₂²×C₃	—	20	2	no	—
286	C₂²×C₃	—	20	2	no	—
287	C₂²×C₃	—	40	4	no	—
288	C₂²×C₃	#4078	1	2	yes	—
289	C₂²×C₃	#4088	1	2	yes	—
290	C₂²×C₃	#4093	1	2	yes	—
291	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	2	no	—
294	C₃×C₄	—	20	2	no	—
295	C₃×C₄	#4093	1	2	yes	—
296	C₃×C₄	C₂×#713	1	2	yes	—
297	Dic₃	—	4	4	no	—
298	C₂×C₅	#5055	1	4	yes	complemented
299	C₂×C₅	C₂×#936	1	1	yes	characteristic
300	C₂×C₅	C₂×#949	1	1	yes	characteristic
301	C₂×C₅	C₂×#951	1	1	yes	characteristic
302	D₅	—	8	2	no	—
303	C₂³	—	10	2	no	—
304	C₂³	—	20	2	no	—
305	C₂³	—	20	2	no	—
306	C₂³	—	20	2	no	—
307	C₂³	C₂×#1201	1	1	yes	characteristic
308	Q₈	—	10	4	no	—
309	Q₈	—	12	8	no	—
310	Q₈	—	20	2	no	—
311	D₄	—	10	4	no	—
312	D₄	—	20	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	6	4	no	—
316	C₂×C₄	—	10	2	no	—
317	C₂×C₄	—	12	4	no	—
318	C₂×C₄	—	20	2	no	—
319	C₂×C₄	#6766	1	4	yes	—
320	C₂×C₄	C₂×#1201	1	2	yes	—
321	C₂×C₄	C₂×#1201	1	2	yes	—
322	C₂×C₄	C₂×#1201	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
324	C₈	—	30	4	no	—
325	C₂×C₃	—	40	2	no	—
326	C₂×C₃	#65378	1	4	yes	complemented
327	C₂×C₃	C₂×#4078	1	1	yes	characteristic
328	C₂×C₃	C₂×#4088	1	1	yes	characteristic
329	C₂×C₃	C₂×#4093	1	1	yes	characteristic
330	C₅	C₂×#5055	1	1	yes	characteristic, complemented, Sylow
331	C₂²	—	20	2	no	—
332	C₂²	—	20	2	no	—
333	C₂²	—	20	2	no	—
334	C₂²	—	40	4	no	—
335	C₂²	#91299	1	2	yes	—
336	C₂²	#91307	1	2	yes	—
337	C₂²	#91368	1	2	yes	—
338	C₂²	C₂²×#1201	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	12	4	no	—
342	C₄	—	20	2	no	—
343	C₄	#91368	1	2	yes	—
344	C₄	C₂×#6766	1	2	yes	—
345	C₃	C₂×#65378	1	1	yes	characteristic, complemented, Sylow
346	C₂	—	40	2	no	—
347	C₂	#11746898	1	4	yes	direct product factor
348	C₂	C₂×#91299	1	1	yes	characteristic
349	C₂	C₂×#91307	1	1	yes	characteristic
350	C₂	C₂×#91368	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1201
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₃×D₂₀	C₂²
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	C₂×#489
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₅×#869	C₂	semidirect
3. C₂×#91087	C₂	semidirect
4. C₂×#90970	C₂	semidirect
6. #11746898	C₂	direct (N × H)
16. C₅×#869	C₂²	semidirect
21. #91087	C₂²	semidirect
22. #90970	C₂²	semidirect
55. C₂×#869	D₅	semidirect
111. #869	C₂×D₅	semidirect
219. C₂×C₃×C₅	#489	semidirect
283. C₃×C₅	C₂×#489	semidirect
298. C₂×C₅	#5055	semidirect
326. C₂×C₃	#65378	semidirect
330. C₅	C₂×#5055	semidirect
345. C₃	C₂×#65378	semidirect
347. C₂	#11746898	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#869	2
C₂×C₃×#4021	2
C₂×#91087	2
C₂×#90970	2
C₂×#90964	2
#11746898	2
C₂×#65378	3
C₂×#5055	5

Subgroups — C2×#11746898 #14437894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746898 #14437894